Problem 56 - College Algebra Essentials

Question

Graph the solution set of each system of inequalities or indicate that the system has no solution. $$\left\\{\begin{array}{l}x^{2}+y^{2} \leq 16 \\\y<2^{x}\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of inequalities is the region which lies inside the circle \(x^{2}+y^{2} \leq 16\) and below the curve \(y<2^{x}\). That is the overlapped region between the two graphs.

Step 1: Understand and graph the inequalities separately

The inequality \(x^{2}+y^{2}\leq16\) depicts the area inside a circle, including its boundary. The center of the circle is the origin (0,0) and has a radius of 4. On the other hand, the graph \(y<2^{x}\) is the region below the graph of the exponential function y=2^x.

Step 2: Identify the intersection of the two regions

The common area, i.e., the solution set of the system, lies in the intersection of the two regions described by the inequalities. Visually, this area represents points that satisfy both inequalities. It's the area inside the circle (and on the circle boundary) that is also below the exponential function.

Step 3: Presenting the final Graph

The graph should highlight the intersection or common region that satisfies the given system of inequalities. This graphically represents all the ordered pairs that are solutions to the original system.

Key Concepts

Solution Set

When dealing with a system of inequalities, such as the given exercise, our task is to find the **solution set**. This is essentially the collection of all points that satisfy both inequalities simultaneously. In mathematical terms, it's a set of ordered pairs

• (x, y) values that make each inequality true.
• This might involve regions that fully or partially overlap.

To determine a solution set for the system of inequalities, it's essential to first understand each part: - The first inequality, \(x^2 + y^2 \leq 16\), forms a circle with radius 4, including its boundary. - The second inequality, \(y < 2^x\), involves an area underneath the exponential curve.

The intersection of these regions, where both conditions are met, gives us the sought solution set.

Graphical Representation

**Graphical representation** helps visualize where solutions to the inequalities lie. By plotting each inequality separately on the same graph, you can easily see where they overlap.

• The circle \(x^2 + y^2 \leq 16\) is shown as a filled area with a boundary of radius 4 from the origin.
• The line \(y = 2^x\) represents the exponential curve, and the inequality \(y < 2^x\) covers everything below this curve.

Visually combining these shows the region of overlap – a distinct area that meets both conditions of the system. This technique supports a deeper understanding of how different mathematical expressions can interact in a dynamic way.

Exponential Function

An **exponential function** like \(y = 2^x\) grows very quickly as \(x\) increases. It has some important properties that help us in graphing and problem-solving:

• This curve crosses the y-axis at (0,1), meaning it never touches or crosses the x-axis.
• As \(x\) gets larger, \(y\) increases rapidly, shooting upwards more steeply.
• The opposite is true as \(x\) becomes negative, where the curve approaches the x-axis, getting closer but never zero.

In this particular problem, the exponential function helps define the boundary of one part of our solution area. Understanding the behavior of exponential functions is key to graphing inequalities involving such curves accurately.

Intersection of Regions

To solve the system of inequalities, finding the **intersection of regions** is crucial. This is the area in the graph where both inequality conditions are simultaneously satisfied.

• This involves overlaying the circle defined by \(x^2 + y^2 \leq 16\) and the area beneath the curve \(y < 2^x\) on the same graph.
• The intersection gives us the **solution set** containing all (x, y) pairs fitting both criteria.

In the context of the problem, the intersection is where the area inside the circle coincides with the area below the exponential curve. Through understanding and finding these intersecting points, students can solve complex systems of inequalities by graphically identifying feasible solutions.